| L(s) = 1 | + (2.77 − 15.7i)2-s + (30.0 + 25.1i)3-s + (−240. − 87.5i)4-s + (−2.54e3 + 925. i)5-s + (480. − 402. i)6-s + (3.96e3 + 6.87e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (−3.15e3 − 1.78e4i)9-s + (7.51e3 + 4.26e4i)10-s + (3.30e4 − 5.71e4i)11-s + (−5.01e3 − 8.68e3i)12-s + (1.08e5 − 9.06e4i)13-s + (1.19e5 − 4.34e4i)14-s + (−9.95e4 − 3.62e4i)15-s + (5.02e4 + 4.21e4i)16-s + (−6.32e4 + 3.58e5i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.213 + 0.179i)3-s + (−0.469 − 0.171i)4-s + (−1.81 + 0.662i)5-s + (0.151 − 0.126i)6-s + (0.624 + 1.08i)7-s + (−0.176 + 0.306i)8-s + (−0.160 − 0.908i)9-s + (0.237 + 1.34i)10-s + (0.679 − 1.17i)11-s + (−0.0697 − 0.120i)12-s + (1.04 − 0.880i)13-s + (0.830 − 0.302i)14-s + (−0.507 − 0.184i)15-s + (0.191 + 0.160i)16-s + (−0.183 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.33170 - 0.770315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33170 - 0.770315i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.77 + 15.7i)T \) |
| 19 | \( 1 + (-5.67e5 + 8.04e3i)T \) |
| good | 3 | \( 1 + (-30.0 - 25.1i)T + (3.41e3 + 1.93e4i)T^{2} \) |
| 5 | \( 1 + (2.54e3 - 925. i)T + (1.49e6 - 1.25e6i)T^{2} \) |
| 7 | \( 1 + (-3.96e3 - 6.87e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-3.30e4 + 5.71e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.08e5 + 9.06e4i)T + (1.84e9 - 1.04e10i)T^{2} \) |
| 17 | \( 1 + (6.32e4 - 3.58e5i)T + (-1.11e11 - 4.05e10i)T^{2} \) |
| 23 | \( 1 + (6.24e5 + 2.27e5i)T + (1.37e12 + 1.15e12i)T^{2} \) |
| 29 | \( 1 + (-5.11e5 - 2.90e6i)T + (-1.36e13 + 4.96e12i)T^{2} \) |
| 31 | \( 1 + (2.09e6 + 3.62e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 2.56e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.46e7 - 1.22e7i)T + (5.68e13 + 3.22e14i)T^{2} \) |
| 43 | \( 1 + (-3.19e7 + 1.16e7i)T + (3.85e14 - 3.23e14i)T^{2} \) |
| 47 | \( 1 + (1.75e6 + 9.98e6i)T + (-1.05e15 + 3.82e14i)T^{2} \) |
| 53 | \( 1 + (7.86e6 + 2.86e6i)T + (2.52e15 + 2.12e15i)T^{2} \) |
| 59 | \( 1 + (-1.46e7 + 8.30e7i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (5.01e6 + 1.82e6i)T + (8.95e15 + 7.51e15i)T^{2} \) |
| 67 | \( 1 + (2.17e7 + 1.23e8i)T + (-2.55e16 + 9.30e15i)T^{2} \) |
| 71 | \( 1 + (1.14e8 - 4.18e7i)T + (3.51e16 - 2.94e16i)T^{2} \) |
| 73 | \( 1 + (-8.31e7 - 6.97e7i)T + (1.02e16 + 5.79e16i)T^{2} \) |
| 79 | \( 1 + (1.22e8 + 1.02e8i)T + (2.08e16 + 1.18e17i)T^{2} \) |
| 83 | \( 1 + (-3.45e8 - 5.99e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-7.87e8 + 6.61e8i)T + (6.08e16 - 3.45e17i)T^{2} \) |
| 97 | \( 1 + (-1.94e8 + 1.10e9i)T + (-7.14e17 - 2.60e17i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.35507021696725593706165829578, −12.45786222580336513791512417292, −11.55672294240196487912247790963, −10.92354553319989187605307264272, −8.854312426698880620323057341889, −8.089822149089994207480288968857, −6.00597016859155656425876878443, −3.88844881881783404315671921818, −3.14575967436362656620186954610, −0.73143411370914858088827103400,
1.04967200525006564013761326027, 3.99477623923642751150333937809, 4.70653261212973566186339643916, 7.26266867911342651494966211012, 7.72450120701480087997323221702, 9.034385516446802884008697993201, 11.12182913366552443878790229177, 12.07087079776688960308106822201, 13.56936671563548117280105925394, 14.48910761629130239231269680319
